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== Wiener–Hopf decomposition ==
The fundamental equation that appears in the Wiener-Hopf method is of the form
The key step in many Wiener–Hopf problems is to decompose an arbitrary function <math>\Phi</math> into two functions <math>\Phi_{\pm}</math> with the desired properties outlined above. In general, this can be done by writing▼
:<math>A(\alpha)\Xi_+(\alpha) + B(\alpha)\Psi_-(\alpha) + C(\alpha) =0, </math>
where <math>A</math>, <math>B</math>, <math>C</math> are known [[holomorphic function]]s, the functions <math>\Xi_+(\alpha)</math>, <math>\Psi_-(\alpha)</math> are unknown and the equation holds in a strip <math>\tau_- < \mathfrak{Im}(\alpha) < \tau_+</math> in the [[Complex_plane|complex <math>\alpha</math> plane]]. Finding <math>\Xi_+(\alpha)</math>, <math>\Psi_-(\alpha)</math> is what's called the '''Wiener-Hopf problem'''.{{sfn | Noble | 1958 | loc=§4.2}}
▲The key step in many Wiener–Hopf problems is to decompose an arbitrary function <math>\Phi</math> into two functions <math>\Phi_{\pm}</math> with the desired properties outlined above.
: <math>\Phi_+(\alpha) = \frac{1}{2\pi i} \int_{C_1} \Phi(z) \frac{dz}{z-\alpha}</math>
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